Constant term

In mathematics, a constant term (sometimes referred to as a free term) is a term in an algebraic expression that does not contain any variables and therefore is constant. For example, in the quadratic polynomial ${\displaystyle x^{2}+2x+3}$, the number 3 is a constant term.

After like terms are combined, an algebraic expression will have at most one constant term. Thus, it is common to speak of the quadratic polynomial ${\displaystyle ax^{2}+bx+c}$, where ${\displaystyle x}$ is the variable, as having a constant term of ${\displaystyle c}$. If the constant term is 0, then it will conventionally be omitted when the quadratic is written out.

Any polynomial written in standard form has a unique constant term, which can be considered a coefficient of ${\displaystyle x^{0}}$. In particular, the constant term will always be the lowest degree term of the polynomial. This also applies to multivariate polynomials. For example, the polynomial ${\displaystyle x^{2}+2xy+y^{2}-2x+2y-4}$ has a constant term of −4, which can be considered to be the coefficient of ${\displaystyle x^{0}y^{0}}$, where the variables are eliminated by being exponentiated to 0 (since any non-zero number exponentiated to 0 becomes 1). For any polynomial, the constant term can be obtained by substituting in 0 instead of each variable; thus, eliminating each variable. The concept of exponentiation to 0 can be applied to power series and other types of series, for example in this power series: $${\displaystyle a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots }$$ ${\displaystyle a_{0}}$ is the constant term.